John of St. Thomas also addresses issues related to the conditions for a valid election

No offense to John of St Thomas, but his analysis is very outdated.  The rules for a conclave and elections have been changed many, many times in the last centuries.  You have to look at the current laws in place for the conclave, since the rules for elections are a disciplinary matter.  Once the election takes place, and the person accepts, then the spiritual aspect comes into play.  Fr Hesse says that the election laws are purely a human/govt matter, which can be changed and have changed over the centuries.

Notice.  Legitimately Elected + Accepted by the Church

Both are required.

Siri was the man who was legitimately elected in 1958.

But if so, he obviously never accepted that election.

I don't want to get derailed by quibbling over whether a layman can be elected.  Let's just say it's a tonsured cleric who is not a priest and not a bishop.

You're claiming that the second he accepts the election, he receives full papal power.  But that's nonsense.  He cannot suddenly exercise Magisterium if he's not even part of the Ecclesia Docens.  He can receive the material aspects of power, such as making appointments, but he cannot formally exercise the office of teaching and ruling.  How can a mere cleric rule over bishop?

He receives the power of jurisdiction immediately.  The power of order would be a separate thing.  A cleric with ordinary jurisdiction could indeed rule over an auxiliary bishop who has no ordinary jurisdiction.

Otherwise, how could Archbishop Lefebvre have allowed Fr. Schmidberger to rule over the bishops of the SSPX?

a man who whimpers away like that and will not stand up for himself in that situation, is most assuredly not going to stand up for Our Lord, or defend or promulgate the faith. If such was the case, then it's most likely that Pope Paul VI was the lesser of two evils.

You have to be kidding.  Paul VI was the lesser of two evils compared to Siri?  I've heard a lot of things, but this takes the cake.  Has your rabid support of R&R made you insane?

Archbishop Lefebvre:

…a grave problem confronts the conscience and the faith of all Catholics since the beginning of Paul VI’s pontificate: how can a pope who is truly successor of Peter, to whom the assistance of the Holy Ghost has been promised, preside over the most radical and far-reaching destruction of the Church ever known, in so short a time, beyond what any heresiarch has ever achieved?

So I'd love to see what Pope Siri would have done.

Fr. Schmidberger's role in the SSPX has nothing to do with jurisdiction.  Leaving aside that the SSPX has no ordinary jurisdiction, religious obedience is not the same thing as jurisdiction.  In point of fact, however, THE reason Archbishop Lefebvre asked that no bishop be the head of the Society is precisely to make the statement that the SSPX is not claiming jurisdiction and therefore setting up schismatic counter-Church.

But the power to exercise Magisterium is not merely an aspect of jurisdiction.  It's part of the TEACHING office of the Church, and only Bishops are part of the Ecclesia Docens.  No mere-cleric Pope could issue an Encyclical to the Universal Church that would have binding force.

Sorry, I think I misunderstood or missed your point in that post that I was responding to.  I understand that in the case where a cleric is elected to the papacy he must seek ordination and/or consecration or he risks losing the office.  But as soon as he accepts the nomination he truly does possess the office and he does exercise ordinary and universal jurisdiction over the whole Church.  But now I think you never said otherwise and I agree with you that he doesn’t exercise Magisterial authority until he is consecrated.  Which is why he is required to seek consecration if he isn’t already a bishop.  And this is why we will probably never see anyone but a bishop nominated.  I remember reading about a cleric appointed to a see who delayed a long time before being ordained and consecrated.  Meanwhile he was exercising his power of jurisdiction.  It was seen as a big scandal.

No offense to John of St Thomas, but his analysis is very outdated.

Not at all. His analysis is clear and perfect. The Church's declaration that a man is Pope is tantamount to the Church's defining of an article of faith. In either case, it is certain and infallible and what the Church has declared must be believed as entirely true. There have been sedevacantists before our times, like William of Ockham and Savonarola. They were in error because they did not understand this principle. You have the benefit of having had it clearly explained by John of St. Thomas.

The rules for a conclave and elections have been changed many, many times in the last centuries.

And the current positive legislation of the Church states that no Cardinal by reason or pretext of any excommunication may be prevented from being elected Roman Pontiff. But beside that, universal acceptance is a guarantee that the election is valid.

You have to look at the current laws in place for the conclave, since the rules for elections are a disciplinary matter.

Pope St. Pius X has explained that as soon as the man elected accepts the elections, he possesses in act, and can exercise, the supreme jurisdiction over the whole world. If there be any doubts in the matter, they can be cleared up by the Church's acceptance of the man elected.

Here is St. Benedict's Centre quoting St. Alphonsus: https://catholicism.org/modern-popes.html

“It doesn’t matter that in past centuries some pontiff has been elected in an illegitimate fashion or has taken possession of the pontificate by fraud: it suffices that he has been accepted after as pope by all the Church, for this fact he has become the true pontiff.”

And Father Gueranger, "“The inevitable play of human passions, interfering in the election of the Vicar of Christ, may perchance for a while render uncertain the transmission of spiritual power. But when it is proved that the Church, still holding, or once more put in possession of, her liberty, acknowledges in the person of a certain Pope, until then doubtful, the true Sovereign Pontiff, this her very recognition is a proof that, from that moment at least, the occupant of the Apostolic See is as such invested by God himself.” (Abbot Guéranger, O.S.B., The Liturgical Year , Vol XII, pg. 1 )

Here is St. Robert on the denial of St. Peter on the morning of Good Friday, "When St. Peter denied Christ, he had not yet begun to be the Supreme Pontiff, for it is certain that Ecclesiastical rule was handed to him by Christ in the last chapter of John, since the Lord said to him after the resurrection: “Simon, son of John, feed my sheep.” Therefore, that denial of Peter cannot be numbered among errors of the Roman Pontiffs. Besides, I add that Peter denied Christ with words, but not truly in his heart: hence Peter did not throw off the confession of faith, nor faith itself, as we showed previously." St. Robert Bellarmine, On the Roman Pontiff, vol. 2, trans. by Ryan Grant [Mediatrix Press, 2016], Book IV, Ch. 8, p. 175) - so see how differently God judges the heart from how man judges the exterior.

I'm sure some sedevacantist wise guy is going to tell me that, in that case, St. Peter was a "pre-election heretic" The Doctors disagree is all I can say.

Jesus Himself has said, "Consider Peter. He denied Me. Why? Not even he knew why. Was Peter a coward? No. My Peter was not cowardly. Facing the cohort and the guards of the Temple he had dared to wound Malcus to defend Me, risking his own life thereby. He then ran away, without the will to do so. Then he denied Me, without the will to do it. Later he did remain and proceed on the bloody way of the Cross, on My Way, until he reached death on a cross. And then he bore witness to Me very efficiently, to the point of being killed because of his fearless faith. I defend My Peter. His bewilderment was the last one of his human nature. But his spiritual will was not present at that moment. Dulled by the weight of his humanity, it was asleep. When it awoke, it did not want to remain in sin, but it wanted to be perfect. I forgave him at once." The Popes may have made mistakes, but God forgave them, and took them to Heaven.

God has judged Pope Paul VI and Pope John Paul II, whatever mistakes they made, to be Saints in Heaven. How will you explain that, Pax Vobis? To me, it is difficult to explain, and the only thing that can come close to the correct answer is what Our Lady is reported to have said at Bayside, that Pope Paul VI is a blessed man and will be martyred, because there was some impostor, and these things were happening against his will. "Your Father, in the eternal city of Rome, Pope Paul VI, your Holy Father, is a blessed man, for he carries his cross. Your Holy Father is a blessed man, for he shall be martyred." - Our Lady at Bayside, June 18, 1977. That came true 41 years later.

I don't know the full explanation. Much of it is still a mystery. But both R&R and SVism fall short of the real explanation, for they are inconsistent. Svism is inconsistent with doctrine, and R&R is inconsistent with itself, in claiming to recognize the Pope while also wanting to be out of communion with him, and not recognizing and venerating the Popes and Saints declared as such ex cathedra by the Pope.

Not at all. His analysis is clear and perfect. The Church's declaration that a man is Pope is tantamount to the Church's defining of an article of faith. In either case, it is certain and infallible and what the Church has declared must be believed as entirely true. There have been sedevacantists before our times, like William of Ockham and Savonarola.

Great, now you can apply the same to Vatican 2 and start attending Novus Ordo instead of illicit Masses in a Society with no canonical status.